for \(\gamma = e^\varepsilon \)  and any pairs of distributions \((P, Q)\)  where \(P{\mathsf {K}}\)  represents the output distribution of \({\mathsf {K}}\)  when the input distribution is \(P\) . Thus, the approximate LDP guarantee of a mechanism can be fully characterized by its contraction under \({\mathsf {E}}_\gamma \) -divergence. When combined with
standard statistical techniques, including Le Cam's and Fano's methods [1]}, [2]}, \({\mathsf {E}}_\gamma \) -contraction leads to general lower bounds for the minimax and Bayesian risk under \((\varepsilon , \delta )\) -LDP for any \(\varepsilon \ge 0\)  and \(\delta \in [0,1]\) . In particular, we show that the price of privacy in this case is to reduce the sample size from \(n\)  to \(n[1-e^{-\varepsilon }(1-\delta )]\) .
